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Journal articles on the topic 'Semigroup stability'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Bodor, Bertalan, Erkko Lehtonen, Thomas Quinn-Gregson, and Nikolaas Verhulst. "HS-stability and complex products in involution semigroups." Semigroup Forum 103, no. 2 (August 10, 2021): 395–413. http://dx.doi.org/10.1007/s00233-021-10213-x.

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AbstractWhen does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular $${}^{*}$$ ∗ -semigroups and commutative involution semigroups.

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2

Chill, R., D. Seifert, and Y. Tomilov. "Semi-uniform stability of operator semigroups and energy decay of damped waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2185 (October 19, 2020): 20190614. http://dx.doi.org/10.1098/rsta.2019.0614.

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Only in the last 15 years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of C 0 -semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems. This article is part of the theme issue ‘Semigroup applications everywhere’.

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3

Han, Xiaoshuang, Mingyan Teng, and Ming Fang. "Well-posedness and Stability of the Repairable System with Three Units and Vacation." Journal of Systems Science and Information 2, no. 1 (February 25, 2014): 54–76. http://dx.doi.org/10.1515/jssi-2014-0054.

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AbstractThe stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroupT(t) generated by the system operator was also shown through the irreducibility of the semigroup.

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4

Xueli, Song, and Peng Jigen. "Equivalence of Lp Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups." Canadian Mathematical Bulletin 55, no. 4 (December 1, 2012): 882–89. http://dx.doi.org/10.4153/cmb-2011-070-0.

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AbstractLp stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is Lp stable for some p > 0. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

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5

Soliman, Ahmed. "A common fixed point theorem for semigroups of nonlinear uniformly continuous mappings with an application to asymptotic stability of nonlinear systems." Filomat 31, no. 7 (2017): 1949–57. http://dx.doi.org/10.2298/fil1707949s.

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In this paper, we study the existence of a common fixed point for uniformly continuous one parameter semigroups of nonlinear self-mappings on a closed convex subset C of a real Banach space X with uniformly normal structure such that the semigroup has a bounded orbit. This result applies, in particular, to the study of an asymptotic stability criterion for a class of semigroup of nonlinear uniformly continuous infinite-dimensional systems.

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6

Acu, Ana Maria, and Ioan Raşa. "A C0-Semigroup of Ulam Unstable Operators." Symmetry 12, no. 11 (November 7, 2020): 1844. http://dx.doi.org/10.3390/sym12111844.

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The Ulam stability of the composition of two Ulam stable operators has been investigated by several authors. Composition of operators is a key concept when speaking about C0-semigroups. Examples of C0-semigroups formed with Ulam stable operators are known. In this paper, we construct a C0-semigroup (Rt)t≥0 on C[0,1] such that for each t>0, Rt is Ulam unstable. Moreover, we compute the central moments of Rt and establish a Voronovskaja-type formula. This enables to prove that C2[0,1] is contained in the domain D(A) of the infinitesimal generator of the semigroup. We raise the problem to fully characterize the domain D(A).

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7

Bobrowski, Adam, and Ryszard Rudnicki. "On convergence and asymptotic behaviour of semigroups of operators." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2185 (October 19, 2020): 20190613. http://dx.doi.org/10.1098/rsta.2019.0613.

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The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed. This article is part of the theme issue ‘Semigroup applications everywhere’.

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8

Braga Barros, Carlos J., Josiney A. Souza, and Victor H. L. Rocha. "Lyapunov stability for semigroup actions." Semigroup Forum 88, no. 1 (October 24, 2013): 227–49. http://dx.doi.org/10.1007/s00233-013-9527-2.

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9

Kumar, Dharmendra, Kalyan B. Sinha, and Sachi Srivastava. "Stability of the Markov (conservativity) property under perturbations." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 02 (June 2020): 2050009. http://dx.doi.org/10.1142/s0219025720500095.

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10

Sun, Xi Ping, Min Luo, and Kai Fang. "A Continuous Semigroup Approach to the Distributional Stability of Nonlinear Models." Applied Mechanics and Materials 525 (February 2014): 653–56. http://dx.doi.org/10.4028/www.scientific.net/amm.525.653.

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We prove the existence of an invariant measure for the continuous semigroup associate with a nonlinear model under the compact set Lyapunov condition. Further,adding the ergodicity of the semigroup operator, we prove the asymptotic stability in distribution for the semigroup. We give a criteria of the asymptotic stability in distribution for the type of evolution equation having a linear generator. Our method is based on continuous semigroup and its generator.We illustrate the result by the Lorenz chaotic model and prove the existence of the natural invariant measure for Lorenz chaotic model.

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11

Braga Barros, Carlos, Victor Rocha, and Josiney Souza. "Lyapunov Stability and Attraction Under Equivariant Maps." Canadian Journal of Mathematics 67, no. 6 (December 1, 2015): 1247–69. http://dx.doi.org/10.4153/cjm-2015-007-7.

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AbstractLetMandNbe admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume thatis a semigroup acting on bothMandN. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors, and Lyapunov stable sets (all concepts deûned for the action of the semigroup) under equivariant maps and semiconjugations fromMtoN.

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12

Acu, Ana Maria, and Ioan Raşa. "Ulam Stability for the Composition of Operators." Symmetry 12, no. 7 (July 13, 2020): 1159. http://dx.doi.org/10.3390/sym12071159.

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Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C 0 -semigroup ( T t ) t ≥ 0 of operators for which each T t is Ulam stable. We present an open problem concerning the Ulam stability of the members of the Bernstein C 0 -semigroup. Two other possible problems are mentioned at the end of the paper.

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13

Birget, Jean-Camille. "Stability and J-depth of expansions." Bulletin of the Australian Mathematical Society 38, no. 1 (August 1988): 41–54. http://dx.doi.org/10.1017/s0004972700027210.

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In this paper I prove that if a semigroup S is stable then ∧L(S) and ∧R(S) (the Rhodes expansions), and ∧+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function.The techniques used were originally developed by the author to prove that ∧+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).

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14

Khodja, Farid Ammar, Assia Benabdallah, and Djamel Teniou. "Stability of coupled systems." Abstract and Applied Analysis 1, no. 3 (1996): 327–40. http://dx.doi.org/10.1155/s1085337596000176.

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The exponential and asymptotic stability are studied for certain coupled systems involving unbounded linear operators and linear infinitesimal semigroup generators. Examples demonstrating the theory are also given from the field of partial differential equations.

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15

Souza, Josiney A., Hélio V. M. Tozatti, and Victor H. L. Rocha. "On stability and controllability for semigroup actions." Topological Methods in Nonlinear Analysis 48, no. 1 (May 9, 2016): 1. http://dx.doi.org/10.12775/tmna.2016.039.

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16

Rezaei, Hamid, and Soon-Mo Jung. "On the stability of metric semigroup homomorphisms." Journal of Mathematical Inequalities, no. 3 (2015): 935–46. http://dx.doi.org/10.7153/jmi-09-76.

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17

Zhang, Caihong, Yinuo Huang, Licheng Wang, Chongxiong Duan, Tiezhu Zhang, and Kai Wang. "Stability of Two Weakly Coupled Elastic Beams with Partially Local Damping." Mathematical Problems in Engineering 2020 (May 6, 2020): 1–9. http://dx.doi.org/10.1155/2020/7169526.

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In this paper, the stability of two weakly coupled elastic beams connected vertically by a spring is investigated via the frequency domain method and the multiplier technique. When the two beams have partially local damping, the operator A is obtained via variable conversion, and it generating a semigroup is proved, then we obtain that the semigroup is exponentially stable by reduction to absurdity.

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18

Dus, Mathias, Francesco Ferrante, and Christophe Prieur. "On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 23. http://dx.doi.org/10.1051/cocv/2019069.

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This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L2([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L∞ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.

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19

Han, Zhong-Jie, and Zhuangyi Liu. "Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 51. http://dx.doi.org/10.1051/cocv/2018060.

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In this paper, the regularity and stability of the semigroup associated with a system of coupled plate equations is considered. Indirect structural or Kelvin-Voigt damping is imposed, i.e., only one equation is directly damped by one of these two damping. By the frequency domain method, we show that the associated semigroup of the system with indirect structural damping is analytic and exponentially stable. However, with the much stronger indirect Kelvin-Voigt damping, we prove that, by the asymptotic spectral analysis, the semigroup is even not differentiable. The exponential stability is still maintained. Finally, some numerical simulations of eigenvalues of the corresponding one-dimensional systems are also given.

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20

Liang, Jin, Falun Huang, and Tijun Xiao. "Exponential stability for abstract linear autonomous functional differential equations with infinite delay." International Journal of Mathematics and Mathematical Sciences 21, no. 2 (1998): 255–59. http://dx.doi.org/10.1155/s0161171298000362.

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Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equationx˙(t)=L(xt) (∗)whereLis a continuous linear operator on some abstract phase spaceBinto a Banach spaceE. We prove that the solution semigroup of(∗)is exponentially stable if and only if the fundamental operator(∗)is exponentially stable and the phase spaceBis an exponentially fading memory space.

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21

Perera, Francesc, Andrew Toms, Stuart White, and Wilhelm Winter. "The Cuntz semigroup and stability of closeC∗-algebras." Analysis & PDE 7, no. 4 (August 27, 2014): 929–52. http://dx.doi.org/10.2140/apde.2014.7.929.

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22

Bianchini, Stefano, and Rinaldo M. Colombo. "On the stability of the standard Riemann semigroup." Proceedings of the American Mathematical Society 130, no. 7 (February 27, 2002): 1961–73. http://dx.doi.org/10.1090/s0002-9939-02-06568-1.

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23

Zakora, D. A. "Exponential Stability of a Certain Semigroup and Applications." Mathematical Notes 103, no. 5-6 (May 2018): 745–60. http://dx.doi.org/10.1134/s0001434618050073.

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24

CHUNG, JAEYOUNG. "GENERAL STABILITY OF THE EXPONENTIAL AND LOBAČEVSKIǏ FUNCTIONAL EQUATIONS." Bulletin of the Australian Mathematical Society 94, no. 2 (March 8, 2016): 278–85. http://dx.doi.org/10.1017/s0004972716000095.

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Let $S$ be a semigroup possibly with no identity and $f:S\rightarrow \mathbb{C}$. We consider the general superstability of the exponential functional equation with a perturbation $\unicode[STIX]{x1D713}$ of mixed variables $$\begin{eqnarray}\displaystyle |f(x+y)-f(x)f(y)|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$ In particular, if $S$ is a uniquely $2$-divisible semigroup with an identity, we obtain the general superstability of Lobačevskiǐ’s functional equation with perturbation $\unicode[STIX]{x1D713}$$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

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25

Kim, Gwang Hui. "Stability of the Pexiderized Lobacevski Equation." Journal of Applied Mathematics 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/540274.

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The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equationf((x+y)/2)2=g(x)h(y), wheref,g,h : G2→ℂare unknown functions on an Abelian semigroup(G,+). The obtained result is a generalization of Gǎvruţa's result in 1994 and Kim's result in 2010.

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26

Ge, Zhao Qiang. "On the Exponential Stability of the Singular Distributed Parameter Systems." Applied Mechanics and Materials 719-720 (January 2015): 496–503. http://dx.doi.org/10.4028/www.scientific.net/amm.719-720.496.

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Exponential stability for the singular distributed parameter systems is discussed in the light of the theory of GE0-semigroup in Hilbert space. The necessary and sufficient conditions concerning the exponential stability are given.

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27

Boutin, Benjamin, and Jean-François Coulombel. "Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (June 20, 2017): 489–519. http://dx.doi.org/10.4208/nmtma.2017.m1525.

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AbstractIn this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

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28

DUFFIELD, N. G., and R. F. WERNER. "MEAN-FIELD DYNAMICAL SEMIGROUPS ON C*-ALGEBRAS." Reviews in Mathematical Physics 04, no. 03 (September 1992): 383–424. http://dx.doi.org/10.1142/s0129055x92000108.

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We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra [Formula: see text] with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of [Formula: see text]. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of [Formula: see text].

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29

Basit, Bolis, and A. J. Pryde. "Ergodicity and stability of orbits of unbounded semigroup representations." Journal of the Australian Mathematical Society 77, no. 2 (October 2004): 209–32. http://dx.doi.org/10.1017/s1446788700013598.

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AbstractWe develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.

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30

Pancheva, E. "Stability of Decomposition in Multiplicative Semigroup of Multidimensional Distributions." Theory of Probability & Its Applications 35, no. 3 (January 1991): 602–4. http://dx.doi.org/10.1137/1135087.

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31

Zumbrun, Kevin, and Peter Howard. "Pointwise semigroup methods and stability of viscous shock waves." Indiana University Mathematics Journal 47, no. 3 (1998): 741–872. http://dx.doi.org/10.1512/iumj.1998.47.1604.

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32

Ma, Zhi Yong. "Polynomial Stability for Timoshenko-Type System with Past History." Applied Mechanics and Materials 623 (August 2014): 78–84. http://dx.doi.org/10.4028/www.scientific.net/amm.623.78.

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In this paper, we consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. We use semigroup method to prove the polynomial stability result with assumptions on past history relaxation function exponentially decaying for the nonequal wave-speed case.

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33

Hu, Ling, Zheng Wu, Zhangzhi Wei, and Lianglong Wang. "Existence and Exponential Stability of Solutions to Stochastic Neutral Functional Differential Equations." Discrete Dynamics in Nature and Society 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/2623035.

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In this paper we consider the existence and stability of solutions to stochastic neutral functional differential equations with finite delays. Under suitable conditions, the existence and exponential stability of solutions were obtained by using the semigroup approach and Banach fixed point theorem.

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34

Liu, Helong, Houbao Xu, Jingyuan Yu, and Guangtian Zhu. "Stability on coupling SIR epidemic model with vaccination." Journal of Applied Mathematics 2005, no. 4 (2005): 301–19. http://dx.doi.org/10.1155/jam.2005.301.

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We develop a mathematical model for the disease which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age. Applying semigroup theory and so forth, we investigate the existence of equilibria. We also discuss local stability of steady states.

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35

Huy, Nguyen, and Pham Bang. "Dichotomy and positivity of neutral equations with nonautonomous past." Applicable Analysis and Discrete Mathematics 8, no. 2 (2014): 224–42. http://dx.doi.org/10.2298/aadm140816015h.

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Consider the linear partial neutral functional differential equations with nonautonomous past of the form (?/?t) F(u(t, ?)) = BFu(t, ?) + ?u(t, ?), t ? 0; (? / ?t) u(t, s) = (? / ?s) u(t, s) + A(s)u(t, s), t ? 0 ? s, where the function u(?, ?) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator ? we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U(t, s))t?s?0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (etB)t?0 on X. Furthermore, under the positivity conditions on (etB)t?0, U, F and ? we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.

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36

Badora, Roman, Roman Ger, and Zsolt Páles. "Additive selections and the stability of the Cauchy functional equation." ANZIAM Journal 44, no. 3 (January 2003): 323–37. http://dx.doi.org/10.1017/s1446181100008051.

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AbstractThe main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

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37

Li, Yingwei. "Pointwise stability of reaction diffusion fronts." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 5 (March 25, 2019): 2216–54. http://dx.doi.org/10.1017/prm.2019.6.

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AbstractUsing pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential convergence in weighted Lp and Sobolev norms, while capturing the new feature of spatial diffusion at Gaussian rate. Novel features of the argument are a pointwise Green function decomposition reconciling spectral decomposition and short-time Nash-Aronson estimates and an instantaneous tracking scheme similar to that used in the study of stability of viscous shock waves.

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38

Jin, Xue-Lian, Yang Zhang, Fu Zheng, and Bao-zhu Guo. "Exponential Stability of the Monotubular Heat Exchanger Equation with Time Delay in Boundary Observation." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/8952647.

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The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.

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39

Clark, Stephen, Yuri Latushkin, Stephen Montgomery-Smith, and Timothy Randolph. "Stability Radius and Internal Versus External Stability in Banach Spaces: An Evolution Semigroup Approach." SIAM Journal on Control and Optimization 38, no. 6 (January 2000): 1757–93. http://dx.doi.org/10.1137/s036301299834212x.

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40

Weiss, George. "Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability." Journal of Differential Equations 76, no. 2 (December 1988): 269–85. http://dx.doi.org/10.1016/0022-0396(88)90075-7.

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41

Soliman, Ahmed H., Dareen N. Ali, and Essam O. Abdel-Rahman. "Analysis and asymptotic stability of uniformly Lipschitzian nonlinear semigroup systems." Journal of the Egyptian Mathematical Society 25, no. 1 (January 2017): 43–47. http://dx.doi.org/10.1016/j.joems.2016.06.003.

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42

Liu, Zhuangyi, and Song Mu Zheng. "Exponential stability of the semigroup associated with a thermoelastic system." Quarterly of Applied Mathematics 51, no. 3 (January 1, 1993): 535–45. http://dx.doi.org/10.1090/qam/1233528.

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43

Toms, Andrew S. "Stability in the Cuntz semigroup of a commutative C*-algebra." Proceedings of the London Mathematical Society 96, no. 1 (July 5, 2007): 1–25. http://dx.doi.org/10.1112/plms/pdm023.

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44

Souza, Josiney A., Carlos J. Braga Barros, and Victor H. L. Rocha. "On attractors and stability for semigroup actions and control systems." Mathematische Nachrichten 289, no. 10 (December 16, 2015): 1272–87. http://dx.doi.org/10.1002/mana.201400389.

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45

Du, Xiankun. "LEFT T-IDEMPOTENCE AND LEFT T-STABILITY OF SEMIGROUP RINGS." Communications in Algebra 29, no. 12 (January 1, 2001): 5477–97. http://dx.doi.org/10.1081/agb-100107940.

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46

Chung, Jaeyoung. "Stability of Pexider Equations on Semigroup with No Neutral Element." Journal of Function Spaces 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/153610.

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LetSbe a commutative semigroup with no neutral element,Ya Banach space, andℂthe set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equationfx+y-gx-h(y)≤ϵfor allx,y∈S, wheref,g,h:S→Y. Using Jung’s theorem we obtain a better bound than that usually obtained. Also, generalizing the result of Baker (1980) we prove the superstability for Pexider-exponential equationft+s-gth(s)≤ϵfor allt,s∈S, wheref,g,h:S→ℂ. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equationft+s=gth(s)for allt,s∈S, a closed form of which is not yet known.

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47

Levan, N. "The left shift semigroup approach to stability of distributed systems." Journal of Mathematical Analysis and Applications 152, no. 2 (November 1990): 354–67. http://dx.doi.org/10.1016/0022-247x(90)90070-v.

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Avalos, George. "The Strong Stability and Instability of a Fluid-Structure Semigroup." Applied Mathematics and Optimization 55, no. 2 (March 2007): 163–84. http://dx.doi.org/10.1007/s00245-006-0884-z.

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Storozhuk, K. V. "Obstructions to the uniform stability of a C 0-semigroup." Siberian Mathematical Journal 51, no. 2 (March 2010): 330–37. http://dx.doi.org/10.1007/s11202-010-0034-3.

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Souza, Josiney A., and Hélio V. M. Tozatti. "Some Aspects of Stability for Semigroup Actions and Control Systems." Journal of Dynamics and Differential Equations 26, no. 3 (June 24, 2014): 631–54. http://dx.doi.org/10.1007/s10884-014-9379-9.

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